Question: You have found the following ages (in years) of all 6 lions at your local zoo: $ 13,\enspace 2,\enspace 1,\enspace 5,\enspace 2,\enspace 7$ What is the average age of the lions at your zoo? What is the variance? You may round your answers to the nearest tenth.
Solution: Because we have data for all 6 lions at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $6$ ages and divide by $6$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\mu} = \dfrac{13 + 2 + 1 + 5 + 2 + 7}{{6}} = {5\text{ years old}} $ Find the squared deviations from the mean for each lion. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $13$ years $8$ years $64$ years $^2$ $2$ years $-3$ years $9$ years $^2$ $1$ year $-4$ years $16$ years $^2$ $5$ years $0$ years $0$ years $^2$ $2$ years $-3$ years $9$ years $^2$ $7$ years $2$ years $4$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{64} + {9} + {16} + {0} + {9} + {4}} {{6}} $ $ {\sigma^2} = \dfrac{{102}}{{6}} = {17\text{ years}^2} $ The average lion at the zoo is 5 years old. The population variance is 17 years $^2$.